For a graph \(G\) , let \(\delta (G)\)  and \(\Delta (G)\)  denote the minimum and maximum degrees of \(G\) , respectively. Let \(V_i(G)\)  be the set of vertices of degree \(i\)  in \(G\) . That is, \(V_i(G) = \lbrace v \in V(G): {\rm deg}(v) = i \rbrace \) . When \(G\)  is clear from context, we simply write \(\Delta \)  for \(\Delta (G)\) , and \(V_i\)  for \(V_i(G)\) . A pendant edge is an edge that is incident to a leaf.
For a vertex \(v \in V(G)\) , adding a pendant edge to \(v\) means adding a new leaf \(v_{0}\)  adjacent to \(v\) .
Similarly, deleting a pendant edge from \(v\), say \(e =vv_{0}\) , means also deleting the leaf \(v_{0}\) . The first inductive property was due to Huang [1]} and Chang et al. [2]}:
